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Properties

Label 2-264-24.11-c1-0-22
Degree $2$
Conductor $264$
Sign $0.948 + 0.315i$
Analytic cond. $2.10805$
Root an. cond. $1.45191$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 − 0.483i)2-s + (−1.69 + 0.348i)3-s + (1.53 − 1.28i)4-s + 2.09·5-s + (−2.08 + 1.28i)6-s + 2.90i·7-s + (1.41 − 2.44i)8-s + (2.75 − 1.18i)9-s + (2.78 − 1.01i)10-s + i·11-s + (−2.15 + 2.71i)12-s − 6.10i·13-s + (1.40 + 3.85i)14-s + (−3.55 + 0.731i)15-s + (0.697 − 3.93i)16-s + 4.60i·17-s + ⋯
L(s)  = 1  + (0.939 − 0.341i)2-s + (−0.979 + 0.201i)3-s + (0.766 − 0.642i)4-s + 0.937·5-s + (−0.851 + 0.524i)6-s + 1.09i·7-s + (0.500 − 0.865i)8-s + (0.918 − 0.394i)9-s + (0.880 − 0.320i)10-s + 0.301i·11-s + (−0.621 + 0.783i)12-s − 1.69i·13-s + (0.374 + 1.03i)14-s + (−0.918 + 0.188i)15-s + (0.174 − 0.984i)16-s + 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.948 + 0.315i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.948 + 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86471 - 0.302258i\)
\(L(\frac12)\) \(\approx\) \(1.86471 - 0.302258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.483i)T \)
3 \( 1 + (1.69 - 0.348i)T \)
11 \( 1 - iT \)
good5 \( 1 - 2.09T + 5T^{2} \)
7 \( 1 - 2.90iT - 7T^{2} \)
13 \( 1 + 6.10iT - 13T^{2} \)
17 \( 1 - 4.60iT - 17T^{2} \)
19 \( 1 - 5.46T + 19T^{2} \)
23 \( 1 + 2.95T + 23T^{2} \)
29 \( 1 + 8.16T + 29T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 - 0.425iT - 37T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + 7.48T + 43T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 + 1.24T + 53T^{2} \)
59 \( 1 + 9.75iT - 59T^{2} \)
61 \( 1 - 0.0675iT - 61T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 + 0.485T + 71T^{2} \)
73 \( 1 - 6.35T + 73T^{2} \)
79 \( 1 + 8.73iT - 79T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 - 6.80T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.99491990092095442987687310886, −11.16074161688311131482262971915, −10.10909031448618053203299692565, −9.624700650212685747665366658382, −7.80102129146212236535784475738, −6.25269103839182434847606423209, −5.70874105056377974974259533709, −5.03216551629120787536049019833, −3.35947853143797647900121672262, −1.77927377905023430920545123037, 1.81085408050001699430051595561, 3.84520529774273288333786277765, 4.97072756657320372232697289650, 5.88310916251120165800676284215, 6.91154677789428182595134551543, 7.44908223380937866573111714911, 9.336107094305808039713842091511, 10.34205772207017384513816928511, 11.42734012270115513380368348931, 11.88063309454752880597452384796

Graph of the $Z$-function along the critical line